The best way to find out if a business name is taken it to Incorporate your business; during this process they will check the name and make you aware if it is already taken.

The reality is if you are starting a cleaning business you need to do this anyway to protect your personal assets. If something were to happen while you are cleaning you are financially responsible. If you incorporate this will protect you personally. Your business can be sued but you can not be personally sued. I would recommend an LLC at the very least.